1.4 Primes

RECTANGLES

The rectangle model of multiplication links a number's geometric structure to its divisors.

Primes are numbers that cannot be represented by rectangles with both dimensions whole numbers greater than one.

Exploring a number's geometric shape leads quite naturally to the notion of prime numbers. Just as before, we can consider a whole number to be a collection of pebbles. We could then address the question of whether it is possible to form a rectangle with the pebbles.

Let's look at a collection of 12 pebbles:

We can arrange these 12 pebbles into various rectangular arrays.

Note that the dimensions of each rectangle—the height and width—multiply to equal 12. We call each of these numbers representing a possible dimension a "divisor" of 12; so 12 has six divisors: 1, 2, 3, 4, 6, and 12. In general, we say that a is a divisor of N if for some whole number b, a × b = N.

Note that any whole number, N, can be represented by a 1 × N rectangle—a single row of pebbles.

Some numbers, such as 12, 15, 20, and 100, can be represented by rectangles that are more interesting than a single row of pebbles.

Other numbers, however, such as 5, 11, 17, and 101, can be represented only by the single-row type of rectangle.

In arithmetic, we call a number "prime" if it has precisely two divisors. These primes are the numbers, such as 2, 3, 5, 7, and 11, whose pebble
representations can be arranged only into the single-row type of rectangle. Numbers with more than two divisors are called "composite." Geometrically, these are numbers that can be represented in dot formations by more than one type of rectangle. Note that the number one, which has precisely one divisor, is considered to be neither prime nor composite.

FACTOR TREES

Factor trees reveal the prime decompositions of composite numbers.

A number that is prime has exactly two divisors, itself and one. If a number is not prime—composite, in other words—we can "factor" it to find all of its constituent prime "factors." For example, the number 30 can be written as 6 × 5, which can in turn be written as a product of all-prime factors: 2 × 3 × 5.

Some numbers have multiple possible factor trees. Let's consider the number 300, for example:

Note, however, that although these factor trees for 300 are different, the set of prime factors generated is the same: 3, 5, 5, 2, and 2.

Any composite number can be decomposed this way into a product of primes. In doing this, we see that primes can indeed be thought of as the "atoms," or fundamental building blocks, of all numbers. Real atoms are the smallest individual pieces of an element, such as gold, that still retain all the properties of that element. In this analogy, a composite number is like a molecule. Breaking apart a molecule generates a collection of atoms of different elements, each of which cannot be broken down further. We perform an analogous breakdown when we decompose a composite number and express its prime decomposition via a factor tree. It is interesting to note that every composite number breaks down into a unique product of primes. How do we know this?

FUNDAMENTAL THEOREM OF ARITHMETIC

The Fundamental Theorem of Arithmetic states that every number has only one prime decomposition.

Primes are the "atoms" of arithmetic.

The fact that every composite number has a unique prime decomposition, a concept known as the fundamental theorem of arithmetic, is often taken for granted. In mathematics we should always be careful to question both our own assumptions and the assumptions of those who would tell us something. Such a questioning attitude derives from the previously mentioned importance of proof, pioneered by the Greek philosophers, logicians, and mathematicians. The tools for proving the fundamental theorem of arithmetic were first laid down by the great mathematician Euclid, who lived in Alexandria in the third century BC. These core concepts were not rigorously expressed, however, until Karl Gauss put them on a solid foundation in his Disquisitiones Arithmeticae, first published in 1801. Expressed in modern language, the fundamental theorem of arithmetic states that:

Every natural number greater than 1 can be written as a product of prime numbers in essentially just one way.

This may seem intuitive, even obvious, but it doesn't have to be the case. We can imagine a number system in which the fundamental theorem of arithmetic does not hold. Take, for example, a set, S, consisting of the numbers {1, 4, 7, 10, 13, 16, 19,..., 3n+1,…}. Each number in this system is one more than a multiple of three. Suppose that these numbers are all we have to work with in performing arithmetic operations in this system. As with the natural numbers, we can have notion of prime and composite in this system—let's call them "S-prime" and "S-composite" respectively.

The number 10 has only two divisors in this system, one and itself. The number 16, on the other hand, has three: 1, 4, and 16. Numbers such as 4, 7, and 10 can be called "S-prime," because they have exactly two divisors within the number system, S. Numbers such as 28 and 16 can be called "S-composite," because they have more than two divisors in S. If S obeys the fundamental theorem of arithmetic, then no matter how we draw a factor tree for an S-composite, we should end up with the same set of S-primes. Is this the case? Let's look at two factor trees of the S-composite number 100:

Notice that 100 can be written as a product of S-primes in two different ways, 4 × 25 and 10 × 10. So, this demonstrates that the fundamental theorem of arithmetic does not hold for our number system S. This suggests to us that we cannot take this seemingly obvious property of natural numbers for granted— hence, Gauss's proof that every natural number greater than one has a unique prime factorization is of great importance in the field of number theory.

Actually, finding a number's prime decomposition—also known as the prime factorization—is relatively straightforward, provided we are aware that our number is divisible by some factor and we know what that factor is. Suppose, however, that we come across some large number and wish to find its prime factors. How would we do this? How would we know whether it even had prime factors other than itself?

If the situation were different, and we wished to multiply two large numbers, our task would be easy—we have good, efficient algorithms for multiplying numbers of any size. Factoring a large number, however, is exceedingly difficult. Most factoring methods involve some version of the "trial and error" strategy. Even for a relatively small number, such as 527, our only real choice is to try dividing it by potential factors until one of them divides it evenly. In this case, it wouldn't take us too long to find that 527 = 17 × 31. For a very large number, however, going through all the possibilities could take an intractably long time.

If someone took two large prime numbers and multiplied them together and then asked us to find the prime factorization of that composite number, we would be in trouble. The number we are given has only two prime factors, and if we don't know either of them, there is no good algorithm for finding them. This "one-way-street" aspect of multiplication and factoring will play a key role in our upcoming discussion of encryption. Before we tackle encryption, however, we should take a closer look at prime numbers, for they themselves hold a great deal of mystery.